Abstract
In this paper we devise a first-order-in-time, second-order-in-space, convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. The unconditional unique solvability, energy stability and stability of the scheme are established. Using the a-priori stabilities, we prove error estimates for our scheme, in both the and norms. The proofs of these estimates are notable for the fact that they do not require point-wise boundedness of the numerical solution, nor a global Lipschitz assumption or cut-off for the nonlinear term. The convergence proof requires no refinement path constraint, while the one involving the norm requires only a mild linear refinement constraint, . The key estimates for the error analyses take full advantage of the unconditional stability of the numerical solution and an interpolation estimate of the form , which we establish for finite difference functions. We conclude the paper with some numerical tests that confirm our theoretical predictions.